• Journal of the Korean Mathematical Society
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• v.50 no.3
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• pp.641-693
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• 2013

We show that the optimistic limits of the colored Jones polynomials of the hyperbolic knots coincide with the optimistic limits of the Kashaev invariants modulo $4{\pi}^2$.

• Bulletin of the Korean Mathematical Society
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• v.55 no.3
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• pp.937-956
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• 2018

This article gives the foundations of the colored Jones polynomial for singular knots. We extend Masbum and Vogel's algorithm [26] to compute the colored Jones polynomial for any singular knot. We also introduce the tail of the colored Jones polynomial of singular knots and use its stability properties to prove a false theta function identity that goes back to Ramanujan.

• Kyungpook Mathematical Journal
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• v.56 no.2
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• pp.639-645
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• 2016

In [3] the main theorem was erroneously stated. We needed to assume that the irrationality measure of 1/r is finite to prove the theorem.

• Journal of the Korean Mathematical Society
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• v.49 no.5
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• pp.993-1015
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• 2012

In this paper, we study the quantum sl($n$) representation category using the web space. Specially, we extend sl($n$) web space for $n{\geq}4$ as generalized Temperley-Lieb algebras. As an application of our study, we find that the HOMFLY polynomial $P_n(q)$ specialized to a one variable polynomial can be computed by a linear expansion with respect to a presentation of the quantum representation category of sl($n$). Moreover, we correct the false conjecture [30] given by Chbili, which addresses the relation between some link polynomials of a periodic link and its factor link such as Alexander polynomial ($n=0$) and Jones polynomial ($n=2$) and prove the corrected conjecture not only for HOMFLY polynomial but also for the colored HOMFLY polynomial specialized to a one variable polynomial.